# nLab quaternion-Kähler manifold

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Definition

A Riemannian manifold $(X,g)$ of dimension $4n$ for $n \geq 2$ is called a quaternion-Kähler manifold if its holonomy group is a subgroup of Sp(n).Sp(1) (where Sp(n) is the $n$th quaternionic unitary group, and in particular $Sp(1) \simeq SU(2) \simeq$ Spin(3), and the central product is the quotient group of the direct product group by the diagonal center $\mathbb{Z}/2$).

If the holonomy group is in fact a subgroup of just the $Sp(n)$-factor, one speaks of a hyperkähler manifold.

Quaternion-Kähler manifolds are necessarily Einstein manifolds (see below). In particular their scalar curvature $R$ is constant, and hence a real number $R \in \mathbb{R}$. If the scalar curvature is positive, then one speaks of a positive quaternion-Kähler manifold.

## Properties

### As part of the Berger classification

classification of special holonomy manifolds by Berger's theorem:

$\,$G-structure$\,$$\,$special holonomy$\,$$\,$dimension$\,$$\,$preserved differential form$\,$
$\,\mathbb{C}\,$$\,$Kähler manifold$\,$$\,$U(n)$\,$$\,2n\,$$\,$Kähler forms $\omega_2\,$
$\,$Calabi-Yau manifold$\,$$\,$SU(n)$\,$$\,2n\,$
$\,\mathbb{H}\,$$\,$quaternionic Kähler manifold$\,$$\,$Sp(n).Sp(1)$\,$$\,4n\,$$\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,$
$\,$hyper-Kähler manifold$\,$$\,$Sp(n)$\,$$\,4n\,$$\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\,$ ($a^2 + b^2 + c^2 = 1$)
$\,\mathbb{O}\,$$\,$Spin(7) manifold$\,$$\,$Spin(7)$\,$$\,$8$\,$$\,$Cayley form$\,$
$\,$G2 manifold$\,$$\,$G2$\,$$\,7\,$$\,$associative 3-form$\,$

### As $\mathbb{H}$-Riemannian manifolds

$\;$normed division algebra$\;$$\;\mathbb{A}\;$$\;$Riemannian $\mathbb{A}$-manifolds$\;$$\;$special Riemannian $\mathbb{A}$-manifolds$\;$
$\;$real numbers$\;$$\;\mathbb{R}\;$$\;$Riemannian manifold$\;$$\;$oriented Riemannian manifold$\;$
$\;$complex numbers$\;$$\;\mathbb{C}\;$$\;$Kähler manifold$\;$$\;$Calabi-Yau manifold$\;$
$\;$quaternions$\;$$\;\mathbb{H}\;$$\;$quaternion-Kähler manifold$\;$$\;$hyperkähler manifold$\;$
$\;$octonions$\;$$\;\mathbb{O}\;$$\;$Spin(7)-manifold$\;$$\;$G2-manifold$\;$

(Leung 02)

### As quaternionic manifolds

###### Example

(quaternion-Kähler manifolds are quaternionic manifolds)

By definition, a quaternion-Kähler manifold $M$ has holonomy group contained in the direct product group Sp(n)$\times$Sp(1), admitting an extension of the Levi-Civita connection $\nabla$ on the holonomy bundle as torsion-free. Thus a quaternion-Kähler manifold is automatically a quaternionic manifold.

Such extension $\nabla_\text{quat}$ of $\nabla$ however is not unique, since $\nabla_\text{quat} + \mathcal{S}$ is another Sp(n)Sp(1)-preserving connection, where $\mathcal{S}$ is a (1, 2)-tensor such that for every $p \in M$, $\mathcal{S}(p)$ takes values in the first prolongation of the Lie algebra for the G-structure.

### Characteristic classes

###### Proposition

Let $X$ be a closed smooth manifold of dimension 8 with Spin structure. If the frame bundle moreover admits G-structure for

$\;\;G =$ Sp(2).Sp(1) $\hookrightarrow$ SO(8)

then the Euler class $\chi$, the second Pontryagin class $p_2$ and the cup product-square $(p_1)^2$ of the first Pontryagin class of the frame bundle/tangent bundle are related by

(1)$8 \chi \;=\; 4 p_2 - (p_1)^2 \,.$
###### Remark

The same conclusion (1) also holds for $Spin(7)$-structure, see there

### Reduction to hyper-Kähler structure

A quaternion-Kähler manifold $(X,g)$ is a hyper-Kähler manifold, hence has $Sp(n) \hookrightarrow Sp(n)\cdot Sp(1)$-structure, precisely if its scalar curvature, which is a constant by $(X,g)$ being an Einstein manifold, vanishes: $R(g) = 0$.

(e.g. Amann 09, below Def. 1.5)

### Positive quaternion-Kähler manifolds

###### Definition

A quaternion-Kähler manifold $(X,g)$ is called positive if

1. it is a geodesically complete

2. its scalar curvature, which is a constant by $(X,g)$ being an Einstein manifold, is a positive number, $R(g) \gt 0$.

(Salamon 82, Section 6, see e.g. Amann 09, Def. 1.5)

###### Proposition

A connected positive quaternion-Kähler manifold (Def. ) is necessarily compact.

###### Proposition

A connected positive quaternion-Kähler manifold (Def. ) is necessarily simply connected.

###### Proposition

For each dimension $dim(X)$ there is a finite number of isometry classes of positive quaternion-Kähler manifolds (Def. ).

###### Proposition

In fact the Wolf spaces are the only known examples of positive quaternion-Kähler manifold (which is not hyper-Kähler ?!), as of today (e.g. Salamon 82, Section 5).

This leads to the conjecture that in every dimension, the Wolf spaces are the only positive quaternion-Kähler manifolds.

The conjecture has been proven for the following dimensions

## Examples

The archetypical example is

This is the first of the list of examples of spaces that are both quaternion-Kähler manifolds as well as symmetric spaces, called Wolf spaces.

See around Prop. above.

Original articles:

Exposition:

Textbook accounts:

• Arthur Besse, Einstein Manifolds, Springer-Verlag 1987.

• Dominic Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000.

In terms of G-structure:

• Edmond Bonan, Sur les $G$-structures de type quaternionien, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 9 (1967) no. 4, p. 389-463 (numdam:CTGDC_1967__9_4_389_0)

• Simon Salamon, Differential Geometry of Quaternionic Manifolds, Annales scientifiques de l’É.N.S. 4e série, tome 19, no 1 (1986), p. 31-55 (numdam:ASENS_1986_4_19_1_31_0)

• Dmitri V. Alekseevsky, Stefano Marchiafava, Quaternionic-like structures on a manifold: Note I. 1-integrability and integrability conditions, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 43-52 (dml:244082)

• Dmitri V. Alekseevsky, Stefano Marchiafava, Quaternionic-like structures on a manifold: Note II. Automorphism groups and their interrelations, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 53-61 (dml:244299)

• Dmitry V. Alekseevsky, Stefano Marchiafava, Quaternionic structures on a manifold and subordinated structures, Annali di Matematica pura ed applicata 171, 205–273 (1996) (doi:10.1007/BF01759388)

• Joseph Thurman, Quaternionic Geometry and Special Holonomy, 2018 (pdf)

Articles discussing quaternion-Kähler holonomy, connection, and relation to other hypercomplex structures:

• Andrei Moroianu, Uwe Semmelmann, Killing Forms on Quaternion-Kähler Manifolds, Annals of Global Analysis and Geometry, November 2005, Volume 28, Issue 4, pp 319–335 (arXiv:math/0403242, doi:10.1007/s10455-005-1147-y)

• Pedersen, Poon, and Swann. “Hypercomplex structures associated to quaternionic manifolds”, Differential Geometry and its Applications (1998) 273-293 North-Holland.

• Misha Verbitsky, Hyperkähler manifolds with torsion, supersymmetry and Hodge theory, Asian J. Math, V. 6 No. 4, pp. 679-712, Dec. 2002.